Resolvent at low energy III: the spectral measure

TL;DR

This paper studies the low-energy spectral measure of the Laplacian on asymptotically conic manifolds, revealing its singularity structure and deriving long-time asymptotics for wave and Schrödinger equations, including Price's law.

Contribution

It characterizes the conormal-Legendrian singularity structure of the spectral measure at low energy on asymptotically conic manifolds and applies this to wave and Schrödinger asymptotics.

Findings

Spectral measure has a conormal-Legendrian singularity structure.

Derived asymptotics for wave operators as time tends to infinity.

Proved Price's law analogue for odd-dimensional asymptotically conic manifolds.

Abstract

Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $\Delta$ be the positive Laplacian associated to $g$, and $P = \Delta + V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(\lambda) = (\lambda/\pi i) \big(R(\lambda+i0) - R(\lambda - i0) \big)$ of $P_+^{1/2}$, where $R(\lambda) = (P - \lambda^2)^{-1}$, as $\lambda \to 0$, in a manner similar to that done previously by the second author and Vasy, and by the first two authors. The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which is obtained from $M^2 \times [0, \lambda_0)$ by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators $\cos(t \sqrt{P_+})$ and $\sin(t \sqrt{P_+})/\sqrt{P_+}$, and the Schr\"odinger propagator $e^{itP}$, as $t \to \infty$. In particular, we prove the analogue of Price's law for odd-dimensional asymptotically conic manifolds. This result on the spectral measure has been used in a follow-up work by the authors (arXiv:1012.3780) to prove sharp restriction and spectral multiplier theorems on asymptotically conic manifolds.